G is the ideal of a birational model of the 5-dimensional quotient F(Q)//(Gm)3 as an anti-canonical hypersurface in a toric variety. The toric variety is a ℙ3-bundle over a ℙ2-bundle over ℙ1. The fibers over the ℙ2-bundle over ℙ1 are K3-surfaces
i1 : G= precomputedCoxModel(QQ)
3 2 3 2 2
o1 = ideal(s s t r r - s t t r r + 2s s t r r r - s s t r r r -
0 2 0 0 2 2 0 1 0 2 0 2 0 0 1 2 0 2 1 0 1 2
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2
2s t t r r r + s t r r r + s r r r - s s r r r + s s t r r r -
2 0 1 0 1 2 2 1 0 1 2 0 0 1 2 0 1 0 1 2 0 2 0 0 1 2
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 3
3s s t r r r + s s t r r r - s t t r r r + 2s t r r r + s r r -
0 2 1 0 1 2 1 2 1 0 1 2 2 0 1 0 1 2 2 1 0 1 2 0 1 2
------------------------------------------------------------------------
3 3 3 2 2 3 2 2 2 2 2 2
s s r r - 2s s t r r + s s t r r + s t r r - s s t r r + s t r r +
0 1 1 2 0 2 1 1 2 1 2 1 1 2 2 1 1 2 0 2 0 0 2 2 0 0 2
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2 2
s t t r r - s s t r r r + s t r r r + s s t r r r - s t r r r -
2 0 1 0 2 0 2 0 0 1 2 2 0 0 1 2 0 2 1 0 1 2 2 1 0 1 2
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2 2 2 2
s r r + s s r r + s s t r r - s s t r r + 2s s t r r - s s t r r -
0 1 2 0 1 1 2 0 2 0 1 2 1 2 0 1 2 0 2 1 1 2 1 2 1 1 2
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 3 3 2 2 3
s t t r r - s t r r - s s t t r r + s s s t t r r + s s t t r r -
2 0 1 1 2 2 1 1 2 0 2 0 1 0 3 0 1 2 0 1 0 3 0 2 0 1 0 3
------------------------------------------------------------------------
2 2 3 3 2 2 2 2 2
s s t t r r + s t r r r - 2s s t r r r + s s t r r r -
1 2 0 1 0 3 0 0 0 1 3 0 1 0 0 1 3 0 1 0 0 1 3
------------------------------------------------------------------------
2 2 2 2 2 2 2 2
2s s t t r r r + 3s s s t t r r r - s s t t r r r + s s t r r r -
0 2 0 1 0 1 3 0 1 2 0 1 0 1 3 1 2 0 1 0 1 3 0 2 1 0 1 3
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 3 2
s s s t r r r + s s t t r r r - s s t t r r r - s s t r r r +
0 1 2 1 0 1 3 0 2 0 1 0 1 3 1 2 0 1 0 1 3 0 2 1 0 1 3
------------------------------------------------------------------------
2 3 2 3 2 2 2 2 2 2 2 2
s s t r r r - s t r r r + 2s s t r r r - s s t r r r + 2s s t r r r
1 2 1 0 1 3 0 1 0 1 3 0 1 1 0 1 3 0 1 1 0 1 3 0 2 1 0 1 3
------------------------------------------------------------------------
2 2 2 2 2 2 3 2 2 3 2
- 3s s s t r r r + s s t r r r - s s t r r r + s s t r r r -
0 1 2 1 0 1 3 1 2 1 0 1 3 0 2 1 0 1 3 1 2 1 0 1 3
------------------------------------------------------------------------
2 2 2 2 2 2 2 2
s s s t r r r + s s t t r r r - 2s s s t t r r r + s s t t r r r -
0 1 2 0 0 2 3 0 2 0 1 0 2 3 0 1 2 0 1 0 2 3 1 2 0 1 0 2 3
------------------------------------------------------------------------
2 2 2 2 2 2 3 2
s s t t r r r + 2s s t t r r r - 2s t r r r r + 3s s t r r r r -
0 2 0 1 0 2 3 1 2 0 1 0 2 3 0 0 0 1 2 3 0 1 0 0 1 2 3
------------------------------------------------------------------------
2 2 2
s s t r r r r - s s s t r r r r + 3s s t t r r r r -
0 1 0 0 1 2 3 0 1 2 0 0 1 2 3 0 2 0 1 0 1 2 3
------------------------------------------------------------------------
2 2 2
4s s s t t r r r r + s s t t r r r r + s s t t r r r r -
0 1 2 0 1 0 1 2 3 1 2 0 1 0 1 2 3 1 2 0 1 0 1 2 3
------------------------------------------------------------------------
2 2 2 2 2 2 2
s s t r r r r + 2s s s t r r r r - s s t t r r r r + s s t t r r r r
0 2 1 0 1 2 3 0 1 2 1 0 1 2 3 0 2 0 1 0 1 2 3 1 2 0 1 0 1 2 3
------------------------------------------------------------------------
2 3 2 3 2 2 2 2
+ s s t r r r r - 2s s t r r r r - 2s s t r r r + 2s s t r r r +
0 2 1 0 1 2 3 1 2 1 0 1 2 3 0 1 0 1 2 3 0 1 0 1 2 3
------------------------------------------------------------------------
3 2 2 2 2 2 2
s t r r r - 3s s t r r r + 2s s t r r r + 3s s s t t r r r -
0 1 1 2 3 0 1 1 1 2 3 0 1 1 1 2 3 0 1 2 0 1 1 2 3
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2
2s s t t r r r - 2s s t r r r + 5s s s t r r r - 2s s t r r r -
1 2 0 1 1 2 3 0 2 1 1 2 3 0 1 2 1 1 2 3 1 2 1 1 2 3
------------------------------------------------------------------------
2 2 2 2 3 2 2 3 2 3 2
s s t t r r r + s s t r r r - 2s s t r r r + s t r r r -
1 2 0 1 1 2 3 0 2 1 1 2 3 1 2 1 1 2 3 0 0 1 2 3
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2
s s t r r r - s s s t r r r + s s t r r r - s s t t r r r +
0 1 0 1 2 3 0 1 2 0 1 2 3 1 2 0 1 2 3 0 2 0 1 1 2 3
------------------------------------------------------------------------
2 2 2 2 2 2 2 2 2 2 3 2 2
s s s t t r r r + s s s t t r r - s s s t t r r - s s s t t r r +
0 1 2 0 1 1 2 3 0 1 2 0 1 0 3 0 1 2 0 1 0 3 0 1 2 0 1 0 3
------------------------------------------------------------------------
2 2 3 2 2 3 2 2 2 2 3 2
s s t t r r - s s t t r r r + 2s s t t r r r - s s t t r r r +
1 2 0 1 0 3 0 1 0 1 0 1 3 0 1 0 1 0 1 3 0 1 0 1 0 1 3
------------------------------------------------------------------------
2 2 2 2 2 2 3 2 2 2 3 2
2s s s t t r r r - 3s s s t t r r r + s s t t r r r - s s s t r r r
0 1 2 0 1 0 1 3 0 1 2 0 1 0 1 3 1 2 0 1 0 1 3 0 1 2 1 0 1 3
------------------------------------------------------------------------
2 3 2 2 3 2 2 2 3 2 2 4 2
+ s s s t r r r - s s s t t r r r + s s t t r r r + s s s t r r r -
0 1 2 1 0 1 3 0 1 2 0 1 0 1 3 1 2 0 1 0 1 3 0 1 2 1 0 1 3
------------------------------------------------------------------------
2 2 4 2 3 2 2 2 2 2 2 2 2 3 2 2 2 2 3 2 2
s s t r r r + s s t r r - 2s s t r r + s s t r r - 2s s s t r r +
1 2 1 0 1 3 0 1 1 1 3 0 1 1 1 3 0 1 1 1 3 0 1 2 1 1 3
------------------------------------------------------------------------
2 3 2 2 3 3 2 2 2 4 2 2 2 2 4 2 2 2 2 2 2
3s s s t r r - s s t r r + s s s t r r - s s t r r + s s t r r r -
0 1 2 1 1 3 1 2 1 1 3 0 1 2 1 1 3 1 2 1 1 3 0 1 0 1 2 3
------------------------------------------------------------------------
3 2 2 3 2 2 2 2 2 2 2
s s t r r r + s s t t r r r - s s t t r r r - s s s t t r r r +
0 1 0 1 2 3 0 1 0 1 1 2 3 0 1 0 1 1 2 3 0 1 2 0 1 1 2 3
------------------------------------------------------------------------
3 2 2 2 2 2 2 2 2
s s t t r r r - s s s t t r r r + s s s t t r r r )
1 2 0 1 1 2 3 0 1 2 0 1 1 2 3 0 1 2 0 1 1 2 3
o1 : Ideal of QQ[s , s , s , t , t , r , r , r , r ]
0 1 2 0 1 0 1 2 3
|
i2 : coxRing = ring G o2 = coxRing o2 : PolynomialRing |
i3 : 5==dim G-3 o3 = true |
i4 : vars coxRing
o4 = | s_0 s_1 s_2 t_0 t_1 r_0 r_1 r_2 r_3 |
1 9
o4 : Matrix coxRing <--- coxRing
|
i5 : degrees coxRing
o5 = {{1, 1, 0}, {1, 1, 0}, {1, 0, 0}, {0, 1, 0}, {0, 1, 0}, {1, 2, 1}, {1,
------------------------------------------------------------------------
2, 1}, {1, 2, 1}, {0, 0, 1}}
o5 : List
|
i6 : degrees source gens G
o6 = {{6, 10, 4}}
o6 : List
|
i7 : sum degrees coxRing
o7 = {6, 10, 4}
o7 : List
|